ORIGAMI

The name Origami comes from the Japanese words ‘ori’ for folding and ‘kami’ for paper.

The aim is to create objects and shapes by folding paper, without using scissors or glue.

Origami as an art comes from Japan and started in the 17th Century.

On this website you can find out how to create beautiful mathematical paper solids.

A **polygon** is a shape which consists only of straight lines. If all angles are the same, we call it a **regular polygon**.

A **polyhedron** is a 3D object which is made up of polygons.

In mathematics, we call the sides of a polyhedron **faces**, we call the lines **edges** and the corners **vertices**.

The mathematician Leonard Euler discovered that in every polyhedron, #Faces + #Vertices

– #Edges = 2.

Platonic Solids are those Polyhedra which consist of only one type of regular polygons and which look the same at every vertex. The Greek scientist and philosopher Plato discovered that there are only five polyhedra of this type. They are named after the Greek name for the number of faces.

Plato belived that the five Platonic Solids correspond to the four ancient Elements, Earth, Water, Air and Fire, as well as the Universe.

Archimedean Solids, like the Platonic ones, consist of regular Polygons and look the same at every corner. However we may use different regular polygons for the faces. Excluding the five Platonic Solids, there are 13 Archimedean Solids (two of which are ‘reflections’).

We can also ‘stellate’ these polyhedra by converting every face into a small pyramid.

This website includes instructions for how to create many of the origami models.

In most cases, normal A4 printing paper (80gsm) is ideal.

It is important to fold extremely accurately and to sharpen creases using your fingernails or the edge of a ruler.

Many models consist of several modules. Fold all modules before assembling them. Usually the last one is the most difficult to connect.

ROBERT LANG

DAVID MITCHELL

MEENAKSHI MUKERJI

SOLIDS

The Tetrahedron has four triangular faces and is the smallest Platonic solid. It has 7 axes of symmetry.

Plato belived it represents the ancient element Fire.

Many molecules have their atoms arranged as a Tetrahedron.

The Cube has six quadratic faces and 13 axes of symmetry.

Plato belived it represents the ancient element Earth.

It is often used for dice. Since it is a “regular” solid, every side has the same probability of landing face up.

The Octahedron has eight triangular faces. It is the “dual” solid of the cube.

Plato belived it represents the ancient element Air.

Many natural crystals are based on an octahedral lattice – including diamond, alum or fluorite.

The Dodecahedron has 12 pentagonal faces and 31 axes of symmetry.

Plato belived that the entire Universe has the shape of a Dodecahedron.

The Icosahedron is the largest Platonic solid and has 20 triangular faces. It is the dual of the Dodecahedron.

Plato belived that it represents the element Water. In fact, many viruses, such as herpes, have icosahedral shells.

SOLIDS

The Truncated Tetrahedron has 8 faces (triangles and hexagons), 12 vertices and 18 edges.

The Cuboctahedron has 14 faces (triangles and squares), 12 vertices and 24 edges.

The Truncated Cube has 14 faces (triangles and octagons), 24 vertices and 36 edges.

The Truncated Octahedron has 14 faces (squares and hexagons), 24 vertices and 36 edges.

The Rhombicuboctahedron has 26 faces (triangles and squares), 24 vertices and 48 edges.

The Truncated Cuboctahedron has 26 faces (4 octagons and 4 triangles), 48 vertices and 72 edges.

The Sub Cube has 38 faces (triangles and squares), 24 vertices and 60 edges.

There are two distinct versions which are reflections of each other.

The Icosidodecahedron has 32 faces (triangles and pentagons), 30 vertices and 60 edges.

The Truncated Dodecahedron has 32 faces (triangles and pentagons), 60 vertices and 90 edges.

The Truncated Icosahedron has 32 faces (pentagons and hexagons), 60 vertices and 90 edges.

It is the shape of a football – the most famous polyhedron in the world. It is also the shape of the Buckminsterfullerene molecule (carbon).

The Rhombicosidodecahedron has 62 faces (triangles, squares and pentagons), 60 vertices and 120 edges.

The Truncated Icosidodecahedron has 62 faces (triangles, hexagons and decagons, i.e. 10-gons), 120 vertices and 180 edges.

The Snub Dodecahedron has 92 faces (triangles and pentagons), 60 vertices and 150 edges.

There are two distinct versions which are reflections of each other.

COMPOUNDS

Five intersecting Tetrahedra form the logo of Mathigon and one of the most beautiful polyhedra. But creating them using Origami is extremely difficult.

The shown models consist of 30 modules each, and 10 pages of A4 paper!

ORIGAMI

These beautiful Origami dragons are easy to fold and look very impressive. They are particularly well-suited for fun mathematics lessons.

BACKGROUND